author | paulson |
Fri, 10 May 2002 22:50:08 +0200 | |
changeset 13134 | bf37a3049251 |
parent 6053 | 8a1059aa01f0 |
child 13149 | 773657d466cb |
permissions | -rw-r--r-- |
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(* Title: ZF/AC.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1994 University of Cambridge |
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The Axiom of Choice |
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This definition comes from Halmos (1960), page 59. |
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*) |
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theory AC = Main: |
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axioms AC: "[| a: A; !!x. x:A ==> (EX y. y:B(x)) |] ==> EX z. z : Pi(A,B)" |
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(*The same as AC, but no premise a \<in> A*) |
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lemma AC_Pi: "[| !!x. x \<in> A ==> (\<exists>y. y \<in> B(x)) |] ==> \<exists>z. z \<in> Pi(A,B)" |
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apply (case_tac "A=0") |
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apply (simp add: Pi_empty1, blast) |
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(*The non-trivial case*) |
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apply (blast intro: AC) |
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done |
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(*Using dtac, this has the advantage of DELETING the universal quantifier*) |
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lemma AC_ball_Pi: "\<forall>x \<in> A. \<exists>y. y \<in> B(x) ==> \<exists>y. y \<in> Pi(A,B)" |
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apply (rule AC_Pi) |
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apply (erule bspec) |
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apply assumption |
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done |
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lemma AC_Pi_Pow: "\<exists>f. f \<in> (\<Pi>X \<in> Pow(C)-{0}. X)" |
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apply (rule_tac B1 = "%x. x" in AC_Pi [THEN exE]) |
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apply (erule_tac [2] exI) |
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apply blast |
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done |
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6053
8a1059aa01f0
new inductive, datatype and primrec packages, etc.
paulson
parents:
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diff
changeset
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lemma AC_func: |
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"[| !!x. x \<in> A ==> (\<exists>y. y \<in> x) |] ==> \<exists>f \<in> A->Union(A). \<forall>x \<in> A. f`x \<in> x" |
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apply (rule_tac B1 = "%x. x" in AC_Pi [THEN exE]) |
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prefer 2 apply (blast dest: apply_type intro: Pi_type) |
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apply (blast intro: elim:); |
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done |
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lemma non_empty_family: "[| 0 \<notin> A; x \<in> A |] ==> \<exists>y. y \<in> x" |
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apply (subgoal_tac "x \<noteq> 0") |
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apply blast+ |
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done |
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6053
8a1059aa01f0
new inductive, datatype and primrec packages, etc.
paulson
parents:
2469
diff
changeset
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lemma AC_func0: "0 \<notin> A ==> \<exists>f \<in> A->Union(A). \<forall>x \<in> A. f`x \<in> x" |
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apply (rule AC_func) |
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apply (simp_all add: non_empty_family) |
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done |
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lemma AC_func_Pow: "\<exists>f \<in> (Pow(C)-{0}) -> C. \<forall>x \<in> Pow(C)-{0}. f`x \<in> x" |
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apply (rule AC_func0 [THEN bexE]) |
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apply (rule_tac [2] bexI) |
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prefer 2 apply (assumption) |
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apply (erule_tac [2] fun_weaken_type) |
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apply blast+ |
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done |
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lemma AC_Pi0: "0 \<notin> A ==> \<exists>f. f \<in> (\<Pi>x \<in> A. x)" |
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apply (rule AC_Pi) |
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apply (simp_all add: non_empty_family) |
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done |
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end |